Faculty Publications, Department of MathematicsCopyright (c) 2014 University of Nebraska - Lincoln All rights reserved.
http://digitalcommons.unl.edu/mathfacpub
Recent documents in Faculty Publications, Department of Mathematicsen-usFri, 14 Nov 2014 09:46:18 PST3600The Time Invariance Principle, Ecological (Non)Chaos, and
A Fundamental Pitfall of Discrete Modeling
http://digitalcommons.unl.edu/mathfacpub/82
http://digitalcommons.unl.edu/mathfacpub/82Fri, 31 Oct 2014 10:50:23 PDT
This paper is to show that most discrete models used for population dynamics in ecology are inherently pathological that their predications cannot be independently verified by experiments because they violate a fundamental principle of physics. The result is used to tackle an on-going controversy regarding ecological chaos. Another implication of the result is that all continuous dynamical systems must be modeled by differential equations. As a result it suggests that researches based on discrete modeling must be closely scrutinized and the teaching of calculus and differential equations must be emphasized for students of biology.
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Bo DengDecimal Spike Code Maximizes Neural Memory Against Retrieval Time
http://digitalcommons.unl.edu/mathfacpub/81
http://digitalcommons.unl.edu/mathfacpub/81Fri, 31 Oct 2014 10:07:27 PDT
For a new class of circuit models for single neurons we demonstrate here that if the circuits’ plastic spike-bursts are used for memory alphabet, then using the first 10 bases tend to maximize the amount of information that can be retrieved from and stored into the circuits in a unit time.
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Bo DengOptimizing A Communication System By Neural Circuits:
The Magic Number 4 And Golden Ratio
http://digitalcommons.unl.edu/mathfacpub/80
http://digitalcommons.unl.edu/mathfacpub/80Fri, 31 Oct 2014 09:57:08 PDT
For a new class of circuit models of neurons we demonstrate here that an artificial but robust and efficient communication system can be constructed using one neural circuit as an encoder/transmitter and another as a receiver/decoder, for which metastable-plastic spike-bursts with the minimal bursting periods are used for its alphabet. We also demonstrate that the spike-bursting alphabet is frequency-modulated onto the encoder’s ion pump current as the signal. We further demonstrate that for this spike-excitation-encoding-decoding (SEED) system, the quadrary alphabet tends to give the maximal transmission rate for all possible sources. We also demonstrate that by a trivial source-to-channel encoding, a binary source with the Golden Ratio distribution can be transmitted at the channel capacity of a binary SEED system.
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Bo DengWhy is the Number of DNA Bases 4?
http://digitalcommons.unl.edu/mathfacpub/79
http://digitalcommons.unl.edu/mathfacpub/79Thu, 16 Oct 2014 13:38:18 PDT
In this paper we construct a mathematical model for DNA replication based on Shannon’s mathematical theory for communication. We treatDNAreplication as a communication channel. We show that the mean replication rate is maximal with four nucleotide bases under the primary assumption that the pairing time of the G–C bases is between 1.65 and 3 times the pairing time of the A–T bases.
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Bo DengThe Origin of 2 Sexes Through Optimization of Recombination Entropy Against Time and Energy
http://digitalcommons.unl.edu/mathfacpub/78
http://digitalcommons.unl.edu/mathfacpub/78Thu, 16 Oct 2014 13:38:16 PDT
Sexual reproduction in nature requires two sexes, which raises the question why the reproductive scheme did not evolve to have three or more sexes. Here we construct a constrained optimization model based on the communication theory to analyze trade-offs among reproductive schemes with arbitrary number of sexes. More sexes on one hand lead to higher reproductive diversity, but on the other hand incur greater cost in time and energy for reproductive success. Our model shows that the two-sexes reproduction scheme maximizes the recombination entropy-to-cost ratio, and hence is the optimal solution to the problem.
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Bo DengNeural spike renormalization. Part I — Universal number 1
http://digitalcommons.unl.edu/mathfacpub/77
http://digitalcommons.unl.edu/mathfacpub/77Thu, 16 Oct 2014 13:38:14 PDT
For a class of circuit models for neurons, it has been shown that the transmembrane electrical potentials in spike bursts have an inverse correlation with the intra-cellular energy conversion: the fewer spikes per burst the more energetic each spike is. Here we demonstrate that as the per-spike energy goes down to zero, a universal constant to the bifurcation of spike-bursts emerges in a similar way as Feigenbaum’s constant does to the period-doubling bifurcation to chaos generation, and the new universal constant is the first natural number 1.
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Bo DengNeural spike renormalization. Part II — Multiversal chaos
http://digitalcommons.unl.edu/mathfacpub/76
http://digitalcommons.unl.edu/mathfacpub/76Thu, 16 Oct 2014 13:38:12 PDT
Reported here for the first time is a chaotic infinite-dimensional system which contains infinitely many copies of every deterministic and stochastic dynamical system of all finite dimensions. The system is the renormalizing operator of spike maps that was used in a previous paper to show that the first natural number 1 is a universal constant in the generation of metastable and plastic spike-bursts of a class of circuit models of neurons.
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Bo DengJOINS AND COVERS IN INVERSE SEMIGROUPS AND
TIGHT C*-ALGEBRAS
http://digitalcommons.unl.edu/mathfacpub/75
http://digitalcommons.unl.edu/mathfacpub/75Thu, 16 Oct 2014 13:38:10 PDT
We show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the C*-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight C*-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph Ʌ, the tight C* -algebra of the inverse semigroup associated to Ʌ is the same as the C*-algebra of Ʌ.
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Allan P. Donsig et al.Gross-Hopkins duality
and the Gorenstein condition
http://digitalcommons.unl.edu/mathfacpub/74
http://digitalcommons.unl.edu/mathfacpub/74Thu, 16 Oct 2014 13:38:08 PDT
Gross and Hopkins have proved that in chromatic stable homotopy, Spanier- Whitehead duality nearly coincides with Brown-Comenetz duality. We give a conceptual interpretation of this phenomenon in terms of a Gorenstein condition [8] for maps of ring spectra.
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W. G. Dwyer et al.Module categories for group algebras
over commutative rings
http://digitalcommons.unl.edu/mathfacpub/73
http://digitalcommons.unl.edu/mathfacpub/73Thu, 16 Oct 2014 13:38:06 PDT
We develop a suitable version of the stable module category of a finite group G over an arbitrary commutative ring k. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented kG-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version of the homotopy category of weakly injective kG-modules and a recollement relating the stable category, the homotopy category, and the derived category of kG-modules.
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Dave Benson et al.Hypergraph Independent Sets
http://digitalcommons.unl.edu/mathfacpub/72
http://digitalcommons.unl.edu/mathfacpub/72Thu, 16 Oct 2014 13:38:04 PDT
The study of extremal problems related to independent sets in hypergraphs is a problem that has generated much interest. There are a variety of types of independent sets in hypergraphs depending on the number of vertices from an independent set allowed in an edge. We say that a subset of vertices is j-independent if its intersection with any edge has size strictly less than j. The Kruskal–Katona theorem implies that in an r-uniform hypergraph with a fixed size and order, the hypergraph with the most r-independent sets is the lexicographic hypergraph. In this paper, we use a hypergraph regularity lemma, along with a technique developed by Loh, Pikhurko and Sudakov, to give an asymptotically best possible upper bound on the number of j-independent sets in an r-uniform hypergraph.
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Jonathan Cutler et al.Ulrich ideals and modules†
http://digitalcommons.unl.edu/mathfacpub/71
http://digitalcommons.unl.edu/mathfacpub/71Thu, 16 Oct 2014 13:38:02 PDT
In this paper we study Ulrich ideals of and Ulrich modules over Cohen–Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and classify Ulrich ideals of numerical semigroup rings and rings of finite CM-representation type.
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Shiro Goto et al.Rings of Frobenius operators
http://digitalcommons.unl.edu/mathfacpub/70
http://digitalcommons.unl.edu/mathfacpub/70Thu, 16 Oct 2014 13:37:59 PDT
Let R be a local ring of prime characteristic. We study the ring of Frobenius operators F(E), where E is the injective hull of the residue field of R. In particular, we examine the finite generation of F(E) over its degree zero component F^{0}(E), and show that F(E) need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of F(E) in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings.
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Mordechai Katzman et al.Metastability And Plasticity In A Conceptual Model of Neurons
http://digitalcommons.unl.edu/mathfacpub/69
http://digitalcommons.unl.edu/mathfacpub/69Fri, 05 Sep 2014 10:41:16 PDT
For a new class of neuron models we demonstrate here that typical membrane action potentials and spike-bursts are only transient states but appear to be asymptotically stable; and yet such metastable states are plastic — being able to dynamically change from one action potential to another with different pulse frequencies and from one spike-burst to another with different spike-per-burst numbers. The pulse and spike-burst frequencies change with individual ions’ pump currents while their corresponding metastable-plastic states maintain the same transmembrane voltage and current profiles in range. It is also demonstrated that the plasticity requires two one-way ion pumps operating in opposite transmembrane directions to materialize, and if only one ion pump is left to operate, the plastic states will be lost to a rigid asymptotically stable state either as a resting potential, or a limit cycle with a fixed pulse frequency, or a spike-burst with a fixed spike-per-burst number. These metastable-plastic pulses and spike-bursts may be used as information-bearing alphabet for a communication system that neurons are thought to be.
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Bo DengConceptual Circuit Models of Neurons
http://digitalcommons.unl.edu/mathfacpub/68
http://digitalcommons.unl.edu/mathfacpub/68Fri, 05 Sep 2014 10:41:14 PDT
A systematic circuit approach tomodel neurons with ion pump is presented here by which the voltage-gated current channels are modeled as conductors, the diffusion-induced current channels are modeled as negative resistors, and the one-way ion pumps are modeled as one-way inductors. The newly synthesized models are different from the type of models based on Hodgkin-Huxley (HH) approach which aggregates the electro, the diffusive, and the pump channels of each ion into one conductance channel. We show that our new models not only recover many known properties of the HH type models but also exhibit some new that cannot be extracted from the latter.
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Bo DengFrom Energy Gradient and Natural Selection to Biodiversity and Stability of Ecosystems
http://digitalcommons.unl.edu/mathfacpub/67
http://digitalcommons.unl.edu/mathfacpub/67Fri, 05 Sep 2014 10:41:12 PDT
The purpose of this paper is to incorporate well-established ecological principles into a foodweb model consisting of four trophic levels --- abiotic resources, plants, herbivores, and carnivores. The underlining principles include Kimura's neutral theory of genetic evolution, Liebig's Law of the Minimum for plant growth, Holling's functionals for herbivore foraging and carnivore predation, the One-Life Rule for all organisms, and Lotka-Volterra's model for intraand interspecific competitions. Numerical simulations of the model led to the following statistical findings: (a) particular foodwebs can give contradicting observations on biodiversity and productivity, in particular, all known functional forms -- - positive, negative, sigmoidal, and unimodal correlations are present in the model; (b) drifting stable equilibria should be expected for ecosystems regardless of their size; (c) resource abundance and specific competitions are the main determining factors for biodiversity, with intraspecific competition enhancing diversity while interspecific competition impeding diversity; (d) endangered species are expected always and loss in lower trophic endangered species are expected at trophication, i.e. the establishment of a higher trophic level of a community. These findings may shed lights on some ongoing debates on biodiversity. In particular, finding (a) implies that the diversity vs. ecosystems functioning debate is most likely the result of incompatible particular observations each cannot be generalized. In particular, general causality should not be expected between diversity and productivity. Finding (b) does not support May's theory that large ecosystems are inherently unstable nor Eton's theory that stability requires diversity. However, it lends a strong support to the energetic theory for the latitudinal diversity gradient. Finding (c) supports Darwin's observation on the effect of interspecific competition on diversity. Finding (d) implies that loss of diversity is inevitable with the appearance of a super species like the human race. Our method and result also suggest that although the evolution of particular species cannot be predicted, some general statistic patterns appear to persist. In addition to the aforementioned findings, these persisting patterns include: the trophic succession, the trophic biomass separation in orders of magnitude, the upper bounds in biodiversity in relationship to the intensities of specific competitions despite the enormous possible number of species allowed by genetic mutations.
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Bo DengThe K-theory of toric varieties in positive characteristic
http://digitalcommons.unl.edu/mathfacpub/66
http://digitalcommons.unl.edu/mathfacpub/66Wed, 26 Feb 2014 07:20:43 PST
We show that if X is a toric scheme over a regular ring containing a field of finite characteristic, then the direct limit of the K-groups of X taken over any infinite sequence of non-trivial dilations is homotopy invariant. This theorem was known in characteristic 0. The affine case of our result was conjectured by Gubeladze.
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G. Cortiñas et al.THREEFOLD FLOPS VIA MATRIX
FACTORIZATION
http://digitalcommons.unl.edu/mathfacpub/65
http://digitalcommons.unl.edu/mathfacpub/65Tue, 19 Nov 2013 07:16:02 PST
The explicit McKay correspondence, as formulated by Gonzalez- Sprinberg and Verdier, associates to each exceptional divisor in the minimal resolution of a rational double point, a matrix factorization of the equation of the rational double point. We study deformations of these matrix factorizations, and show that they exist over an appropriate “partially resolved” deformation space for rational double points of types A and D. As a consequence, all simple flops of lengths 1 and 2 can be described in terms of blowups defined from matrix factorizations. We also formulate conjectures which would extend these results to rational double points of type E and simple flops of length greater than 2.
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Carina Curto et al.DG algebras with exterior homology
http://digitalcommons.unl.edu/mathfacpub/64
http://digitalcommons.unl.edu/mathfacpub/64Tue, 19 Nov 2013 06:53:16 PST
We study differential graded algebras (DGAs) whose homology is an exterior algebra over a commutative ring R on a generator of degree n, and also certain types of differential modules over these DGAs. We obtain a complete classification with R = Z or R = F_{p}and n ≥ –1. The examples are unexpectedly interesting.
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W. G. Dwyer et al.Encoding Binary Neural Codes in Networks
of Threshold-Linear Neurons
http://digitalcommons.unl.edu/mathfacpub/63
http://digitalcommons.unl.edu/mathfacpub/63Wed, 06 Nov 2013 10:11:35 PST
Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still poorly understood. Here we consider this problem for networks of threshold-linear neurons whose computational function is to learn and store a set of binary patterns (e.g., a neural code) as “permitted sets” of the network. We introduce a simple encoding rule that selectively turns “on” synapses between neurons that coappear in one or more patterns. The rule uses synapses that are binary, in the sense of having only two states (“on” or “off”), but also heterogeneous, with weights drawn from an underlying synaptic strength matrix S. Our main results precisely describe the stored patterns that result from the encoding rule, including unintended “spurious” states, and give an explicit characterization of the dependence on S. In particular, we find that binary patterns are successfully stored in these networks when the excitatory connections between neurons are geometrically balanced—i.e., they satisfy a set of geometric constraints. Furthermore, we find that certain types of neural codes are natural in the context of these networks, meaning that the full code can be accurately learned from a highly undersampled set of patterns. Interestingly, many commonly observed neural codes in cortical and hippocampal areas are natural in this sense. As an application, we construct networks that encode hippocampal place field codes nearly exactly, following presentation of only a small fraction of patterns. To obtain our results, we prove new theorems using classical ideas from convex and distance geometry, such as Cayley-Menger determinants, revealing a novel connection between these areas of mathematics and coding properties of neural networks.
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Carina Curto et al.